Hedge fund Elliott Management’s big bet on Telecom Italia is misfiring badly, but its increasingly bitter rival for control of the Italian telecommunications company is losing even more. It is hard to see progress at Telecom Italia before the two big shareholders stop squabbling.

Media conglomerate Vivendi, Telecom Italia’s biggest shareholder, has been battling Elliott ever since the hedge fund disclosed a stake in March. Telecom Italia’s shares have become collateral damage. They had another dismal day Friday after Italy’s...